• Motivation (Why and How?)
• What are Distance Maps?
• Skeletons
  • Tortuosity
• What are thickness maps?
  • Thickness with Skeletons

• Watershed Segmentation
  • Connected Objects
• Curvature
  • Characteristic Shapes

To simplify our data, but an ellipse model is too simple for many shapes

So while bounding box and ellipse-based models are useful for many object and cells, they do a very poor job with the sample below.

Why

• We assume an entity consists of connected pixels (wrong)
• We assume the objects are well modeled by an ellipse (also wrong)

What to do?

• Is it 3 connected objects which should all be analzed seperately?
• If we could divide it, we could then analyze each spart as an ellipse
• Is it one network of objects and we want to know about the constrictions?
• Is it a cell or organelle with docking sites for cell?
• Neither extents nor anisotropy are very meaningful, we need a more specific metric which can characterize

The distance map is an image where each voxel contains a value. For each image there are two possible distance maps.

1. Foreground where each voxel in the foreground becomes a value for the distance to the closest background voxel.
2. Background where each voxel in the background becomes a value for the distance to the closest foreground voxel.

One of the most useful components of the distance map is that it is relatively insensitive to small changes in connectivity.

• Component Labeling would find radically different results for these two images
  • One has 4 small circles
  • One has 1 big blob

Foreground

Background

For some structures like cellular materials and trabecular bone, we want a more detailed analysis than just thickness. We want to know

• which structures are connected
• how they are connected
• express the network in a simple manner
  • quantify tortuosity
  • branching

The first step is to take the distance transform the structure \[ I_d(x,y) = \textrm{dist}(I(x,y)) \] We can see in this image there are already local maxima that form a sort of backbone which closely maps to what we are interested in.

By using the laplacian filter as an approximate for the derivative operator which finds the values which high local gradients.

\[ \nabla I_{d}(x,y) = (\frac{\delta^2}{\delta x^2}+\frac{\delta^2}{\delta y^2})I_d \approx \underbrace{\begin{bmatrix} -1 & -1 & -1 \\ -1 & 8 & -1 \\ -1 & -1 & -1 \end{bmatrix}}_{\textrm{Laplacian Kernel}} \otimes I_d(x,y) \]

We can locate the local maxima of the structure by setting a minimum surface distance \[ I_d(x,y)>MIN-SLOPE \] and combining it with a minimum slope value \[ \nabla I_d(x,y) > MIN-DIST \]

Thresholds

Harking back to our first lectures, this can be seen as a 2D threshold of the entire dataset.

• We first make the dataset into a tuple

\[ \textrm{cImg}(x,y) = \langle \underbrace{I_d(x,y)}_1, \underbrace{\nabla I_d(x,y)}_2 \rangle \]

\[ \textrm{skelImage}(x,y) = \] \[ \begin{cases} 1, & \textrm{cImg}_1(x,y)\geq MIN-DIST \\ \textbf{ AND} & \textrm{cImg}_2(x,y)\geq MIN-SLOPE \\ 0, & \textrm{otherwise} \end{cases} \]

Pruning

• The structure is a overgrown
• Stricter 'thresholds'

Calculating the distance map by drawing a sphere at each point is very time consuming (\( O(n^3) \)).

• The skeleton (last section) is very closely related to the thickness.
• We found the local maxima in the image using the Laplace
• We can thus grow the Spheres from these points instead of all
• Start by instead of thresholding transforming the image to the distance at each point

\[ \textrm{thSkelImage}(x,y) = \] \[ \begin{cases} \textrm{cImg}_1(x,y) , & \textrm{cImg}_1(x,y)\geq MIN-DIST \\ & \& \textrm{ cImg}_2(x,y)\geq MIN-SLOPE \\ 0, & \textrm{otherwise} \end{cases} \]

   user  system elapsed 
  5.057   0.092   5.224 

   user  system elapsed 
 17.439   0.499  18.637 

It depends

• Small structures are lost
• They might not have been very important or noisy anyways
• Higher values are very similar

   user  system elapsed 
  2.643   0.045   2.742